At the World's End

One hobby of mine is giving people with heterodox ideas the time of day. I was never a flat-earther, but when I was younger, I did have quite a few heterodox beliefs that I now consider to be ridiculous. Contrary to what one might suspect however, every one of those beliefs was eventually overturned by one or two very simple arguments, always delivered by someone who took the time to understand the claims and faithfully point out the errors. So I don’t buy the commonly held idea that arguing with such people is fruitless. In my experience, it’s more often people with orthodox views who are less willing to examine their own beliefs.

I once found myself in the precarious position of feeling like I needed to defend flat-earthers (not because the earth is flat), but because the debunkings on offer were themselves false. This article was born from the conversation that took place. The idea that I’ll be defending is that, from the perspective of a standing-height observer, in most normal conditions, what is seen in the distance is basically the same whether standing on a flat plane, or an earth-sized sphere. The horizon would appear basically the same and the way things disappear into the distance would also appear basically the same on both worlds.

A Flat World’s Horizon

Every point on the ground, no matter how far away, projects into your visual field somewhere below eye level. As distance grows, those points pile up closer and closer to eye level without ever crossing it.

That asymptote is a horizon: a sharp, straight boundary where an infinite amount of ground gets compressed into an infinitely thin band. Projective geometry is the mathematical field that studies how 3D and 2D geometry “project” onto one another. Experts in this field call this phenomenon the vanishing line of a plane (the standard treatment is Hartley & Zisserman’s Multiple View Geometry in Computer Vision). An infinite flat plane does not give you an open, edgeless view. It gives you exactly half ground, half sky, split at eye level with an obvious distinctive line.

On a sphere, a second effect is also at play. The surface falls away from you. Your sight-line to the horizon is the tangent line that grazes the curve, and that tangent points below the horizontal. Because of this, there is a distinctive difference that could be noticed by someone with a good leveling device, a very flat unobstructed landscape, and a very tall building to stand on. But to the naked eye, these two worlds would look basically identical.

The Observable Difference

At eye level, these views are indistinguishable. You can see in the visualization below that even at airliner altitude, the difference is mostly indistinguishable by the naked eye. But the visualization also makes it clear what the distinct difference would actually be.

On the plane, the horizon is welded to eye level. No matter your altitude, the line stays dead center in your field of view. On a sphere, the horizon dips below eye level by an angle of roughly √(2h/R), growing with altitude.

From an airliner the dip is about 3.4°. In practice, atmospheric refraction lifts the horizon slightly, shaving roughly a sixth off the geometric dip. Either way: very subtle to the naked eye, but measurable with an inclinometer.

Atmospheric Gradient

On an infinite plane, a sightline aimed just below eye level passes through an infinitely deep column of air. Scattering, the same aerial perspective that turns distant mountains pale blue, makes it so the farthest ground you see would fade toward sky color. See Minnaert’s Light and Color in the Outdoors documenting these effects. An observer on a flat world is definitely subject to this effect, but there’s a catch.

The same projection geometry that creates the vanishing line also hides the fade. At standing eye height, ground 6 km away already sits just one arcminute below eye level. Everything from 6 km to infinity, including the entire haze gradient, is packed into a band thinner than your eye can resolve. The gradient is real, but you wouldn’t be able to see it. At standing height, the haze gradient would still appear as a distinct line.

At standing height, ground 100 m away sits a full degree below eye level; 1 km away, six arcminutes; and everything from 6 km to infinity is crushed between one arcminute and zero. The haze needs distance to work: on a 40 km-visibility day, ground 1 km away still delivers about 91% of its light, 6 km away 56%, and 40 km away just 2%. So the fade from “ground” to “sky color” plays out between roughly 6 and 60 km, entirely inside the sliver. The light arriving at your eye does form a gradient, but all of it lands on a single photoreceptor’s height on your retina.

This is also why distant mountains do look hazy. A 2,000 m ridge 40 km away sticks up about 3° into the sky, hundreds of arcminutes for the pale blue veil to spread across. Flat ground is the special case: it approaches eye level asymptotically, so perspective crushes all of its far distance into the sliver.

On the sphere, the horizon at standing height is only about 4.7 km away, and five kilometers of clear air scatters barely enough light to tint it. Sharp there too. Near the surface, the atmosphere doesn’t break the tie; it just adds the same faint blue veil to both worlds.

Altitude is where the effect becomes observable. Climb high enough, and the flat world’s haze band widens. Roughly, your altitude divided by the visibility range in radians. On the sphere, the band drifts downward with the dip, which is obscured by the haze, plus or minus relative to how clear a day it is.

Disappearing into the Distance

What sets an object’s apparent size is its angular size, which is the slice of your visual field that it subtends. The math is roughly (physical size ÷ distance). That’s projection geometry, and projection geometry is the primary explanation. The physics of light is involved the way gyroscopic stability is involved in riding a bicycle: technically present, but doing almost nothing to explain the phenomenon.

So what is happening?

As objects move away from an observer in 3-dimensional space, the 2-dimensional geometry that makes up the observer’s field of view shrinks the object. Once its angular size drops below your eye’s resolution limit (at ~1 arcminute, the definition of 20/20 vision, a 1.7 m tall person would hit the limit near 5.8 km), it stops being an extended object and becomes a point. What happens after that point depends what’s behind it.

Against a daytime background, photon flux would have no noticeable effect. A diffusely reflecting person can never appear brighter than they do up close, and up close they’re about as bright as every other sunlit surface around them. So when they shrink below your resolution limit, their light doesn’t vanish, it gets smeared across your eye’s blur spot and averaged with the background directly behind them. Their share of that blur spot falls as 1/d², and their contrast against the scene collapses. They don’t fade to black; they fade into the scenery. Your eye isn’t starved of photons, it’s drowning in them. The limit is contrast. Meteorologists define visibility as the distance at which contrast falls below a threshold, a relation known as Koschmieder’s law, and the classic book-length treatment is W. E. K. Middleton’s Vision Through the Atmosphere (1952). On the idealized plane, your distant friend disappears twice, both times by geometry: first shrunk below resolution, then diluted below contrast.

The simulation below does not simulate the physics of light, it is just simple 3d geometry projected on to a 2d viewport. With naked-eye observation, they are indistinguishable. Even at binocular magnification they are almost indistinguishable. To see any difference at all, for most objects that are small enough to move, you’d need around 60× magnification, about what you can get with a spotting scope. The difference between the two worlds becomes quite obvious once you’re at +100x magnification.

Where Light Matters

Against a dark background, reflected light can carry a sub-resolution object to visibility. This is where photometry starts to have some effect. Stars are the familiar example, but satellites are the sharper one because they are diffuse reflectors (they don’t produce their own light, they scatter sunlight). Sunlit metal and solar panels, far below your eye’s resolution limit, yet they are visible at night because they’re the only bright thing against a dark sky.

This can also have an observable effect in daylight due to specular reflection. A diffuse surface scatters sunlight into a full hemisphere, capping its surface brightness at “ordinary sunlit object.” A mirror redirects an image of the sun itself, tens of thousands of times brighter than a sunlit landscape. This is how a signal mirror works. The 1894 heliograph record between mountain peaks in Utah and Colorado was 183 miles, from mirrors eight inches across, far below resolution at that range.

So photometry does play some role when trying to see things at a distance, but the effect that it plays is not noticeable by a normal human observer, in normal conditions, against normal objects.

Climbing Higher, Seeing Further

Everyone knows that climbing higher lets you see further, and it’s tempting to hear that as a fact about the shape of the world. But most of what you notice when you climb is, once again, projective geometry. At standing height, everything beyond a few kilometers is crushed into the thin band just below eye level. Climb, and it stretches: ground and objects that were compressed into the sliver stretch vertically apart in your visual field. Distant things appear to “come into view,” but they were in view the whole time. They aren’t being revealed; they’re being visually uncompressed.

You can watch this directly in the rig below. Leave the altitude at 2 m and the mountain at 20 km, turn the atmosphere off so nothing but geometry remains, and drag the altitude slider up. The mountain and the ground beyond it unfold the same way on both worlds. The difference between them is basically indistinguishable.

Now set the altitude back to 2 m and drag the mountain distance instead. The mountain shrinks and squashes down into the horizon line identically on both worlds, because the compression into the perspective horizon doesn’t care about the shape of the surface underneath. You won’t see the worlds appear any different until the mountain is about 80 km out, by which point the sphere has quietly swallowed a few hundred meters of its base. That is the one effect the flat plane can’t reproduce, and to the naked eye it’s barely there. Turn the atmosphere back on and the difference between the worlds almost entirely disappears.

What is the Point of This

If you’re going to be effective at persuading others on matters for which there are good arguments and scientific/mathematical modeling, you actually have to be correct about the details. You have to be able to point out what would be the actual observed consequences if the other person’s claim were true, and to figure that out, you have to understand what that person’s argument and perspective is.

In 1870, Alfred Russel Wallace, co-discoverer of natural selection, accepted a flat-earther’s wager and settled it with a careful measurement along the Bedford Level. He got every detail correct and was proven right. What he didn’t get was a concession. What he got instead was years of ridicule for being right. In the end, being correct about the details is the only part we can control. It’s also the only part.